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Chemical Diffusion

Chemical diffusion happens due to a chemical gradient, that is, a difference in concentration of a given contaminant between two points. The diffusion happens from the point with higher concentration towards the point with lower concentration. The mass flux (\(F_D\)) is governed by the following equation, known as Fick’s first law:

\(F_D=-D^{*}n\frac{dC}{dx}\)
\(D^{*}\) = diffusion coefficient
\(\frac{dC}{dx}\) = chemical gradient
\(n\) = porosity

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Chemical Advection

If there is flow of the liquid, the contaminant flows with it. The liquid makes the chemical front move, so the migration is caused by a hydraulic gradient. In order to determine the contaminant migration due to advection only, the equation below can be used:

\(F_A=nv_sC=-k\frac{dh}{dx}C\)
\(F_A\) = mass flux due to advection only
\(C\) = contaminant concentration
\(v_s\) seepage velocity
\(k\) = hydraulic conductivity
\(\frac{dh}{dx}\) = hydraulic gradient

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Mechanical Dispersion

The speed of the flow is not constant in a given plane perpendicular to the direction of the flow, rather it varies locally, because of interactions between the liquid and the solid particles. The equation below describes the mass fulx caused by mechanicald dispersion:

\(F_m=-D_mn\frac{dC}{dx}\)
\(D_m=\alpha _L v_s\)
\(F_m=-\alpha _L v_sn\frac{dC}{dx}\)
\(D_m\) = dispersion coefficient
\(\alpha _L\) = longitudinal dispersivity

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Combined Transport

\(F_T=F_D+F_A+F_m\)
\(F_T=nv_sC-D^{*}n\frac{dC}{dx}-D_mn\frac{dC}{dx}\)
\(D_h=D^{*}+D_m=D^{*}+v_s\alpha _L\)
\(F_T=nv_sC-D_hn\frac{dC}{dx}\)

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Differential Equation controling the non-reactive contaminant transport

\(D_h\frac{d^2C}{dx^2}-v_s\frac{dC}{dx}=\frac{dC}{dt}\)

Solution: \(C(x,t)\)

Boundary and initial conditions:

Solution (Ogata 1970):

\(\frac{C(x,t)}{C_0}=\frac{1}{2}\left[erfc\left(\frac{x-v_st}{2\sqrt{D_ht}}\right)+\exp\left(\frac{v_sx}{D_h}\right)erfc\left(\frac{x+v_st}{2\sqrt{D_ht}}\right)\right]\)

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Differential Equation controling the reactive contaminant transport

\(\frac{D_h}{R_d}\frac{d^2C}{dx^2}-\frac{v_s}{R_d}\frac{dC}{dx}=\frac{dC}{dt}\)

Solution: \(C(x,t)\)

Boundary and initial conditions:

Solution (Ogata 1970):

\(\frac{C(x,t)}{C_0}=\frac{1}{2}\left[erfc\left(\frac{R_dx-v_st}{2\sqrt{R_dD_ht}}\right)+\exp\left(\frac{v_sx}{D_h}\right)erfc\left(\frac{R_dx+v_st}{2\sqrt{R_dD_ht}}\right)\right]\)

Error Function

Advection-Dispersion Profile

Variable Value
Water head 30.48 cm (1 ft)
Soil Thickness 91.44 cm (3 ft)
Hydrodynamic dispersion coefficient 6*10^(-6) cm²/s
Hydraulic conductivity 5*10^(-8) cm/s
Porosity 0.5
Retardation factor variable
Time variable

15 Years

R = 1

R = 2

R = 0.8

22 Years

R = 1

R = 2

R = 0.8

50 Years

R = 1

R = 2

R = 0.8

Breakthrough curves

Variable Value
Water head 30.48 cm (1 ft)
Soil Thickness 91.44 cm (3 ft)
Hydrodynamic dispersion coefficient 6*10^(-6) cm²/s
Hydraulic conductivity 5*10^(-8) cm/s
Porosity 0.5
Retardation factor 1
Time variable

Transit time

Variable Value
Water head 30.48 cm (1 ft)
Soil Thickness 91.44 cm (3 ft)
Hydrodynamic dispersion coefficient 6*10^(-6) cm²/s
Porosity 0.5
Retardation factor 1
Hydraulic conductivity varaible
Time variable